#9390456

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(2,4)(3,9)(8,12)(10,15)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(1,2,6,4)(3,7,9,11)(5,12,13,8)(10,16,15,14)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(3,9)(7,11)(10,15)(14,16)(18,20)(19,26)(25,28)(27,32)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,38,39,46)(34,41,36,44)(35,43,37,45)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9297652);